Quasi-kernels in split graphs

TL;DR

The paper investigates quasi-kernels in split digraphs, addressing the small quasi-kernel conjecture within this structured graph family. It proves a $\frac{2}{3}$ upper bound on the minimum quasi-kernel size for sink-free split digraphs and strengthens this by showing a bound of at most two for sink-free biorientations of complete split graphs, with sinks determining the minimum when present. The authors also establish a $\W[2]$-hardness result for computing a minimum-size quasi-kernel in split digraphs and demonstrate equivalences between sink-free and non-sink-free formulations for $\alpha \ge \frac{1}{2}$, along with complete-split graph observations. Together, these results delineate both the algorithmic complexity and tight structural bounds for quasi-kernels in split graphs, highlighting both promising special-case algorithms and intrinsic hardness in general. The work advances understanding of quasi-kernel sizes, linking combinatorial constructions to parameterized complexity within a meaningful graph class.

Abstract

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The ``small quasi-kernel conjecture,'' proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a $1/2$ ratio, but even with larger ratio, this property is known to hold only for few classes of graphs. The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph $D$ has a quasi-kernel of size at most $\frac{2}{3}|V(D)|$, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

Theorems & Definitions (28)

• Conjecture 1: "Small quasi-kernel conjecture"
• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof